Collapsibility and vanishing of top homology in ... - Technion math
Collapsibility and vanishing of top homology in random simplicial complexes L. Aronshtam∗
N. Linial†
T. Luczak‡
R. Meshulam§
November 2, 2012
Abstract Let ∆n−1 denote the (n − 1)-dimensional simplex. Let Y be a random d-dimensional subcomplex of ∆n−1 obtained by starting with the full (d − 1)-dimensional skeleton of ∆n−1 and then adding each d-simplex independently with probability p = nc . We compute an explicit constant γd , with γ2 ' 2.45, γ3 ' 3.5 and γd = Θ(log d) as d → ∞, so that for c < γd such a random simplicial complex either collapses to a (d − 1)-dimensional subcomplex or it contains ∂∆d+1 , the boundary of a (d + 1)-dimensional simplex. We conjecture this bound to be sharp. In addition we show that there exists a constant γd < cd < d + 1 such that for any c > cd and a fixed field F, asymptotically almost surely Hd (Y ; F) 6= 0.
1
Introduction
Let G(n, p) denote the probability space of graphs on the vertex set [n] = {1, . . . , n} with independent edge probabilities p. It is well known (see e.g. ∗
Department of Computer Science, Hebrew University, Jerusalem 91904, Israel. e-mail: [email protected] . † Department of Computer Science, Hebrew University, Jerusalem 91904, Israel. e-mail: [email protected] . Supported by ISF and BSF grants. ‡ Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Pozna´ n, Poland. e-mail: [email protected] . Supported by the Foundation for Polish Science. § Department of Mathematics, Technion, Haifa 32000, Israel. e-mail: [email protected] . Supported by ISF and ESF grants.
1
[2]) that if c ≥ 1 then a graph G ∈ G(n, nc ) a.a.s. contains a cycle, while for a constant c < 1 √ c c2 c (1) lim Pr [ G ∈ G(n, ) : G acyclic ] = 1 − c · exp( + ). n→∞ n 2 4 In this paper we consider the analogous question for d-dimensional random complexes. There are two natural extensions to the notion of an acyclic graph. Namely, the vanishing of the d-th homology, and collapsibility to a (d − 1)-dimensional subcomplex. These are the two questions we consider here. We provide an upper bound on the threshold for the vanishing of the d-th homology and a lower bound (which we believe to be tight) for the threshold for collapsibility. For a simplicial complex Y , let Y (i) denote the i-dimensional skeleton of Y . Let Y (i) be the set of i-dimensional simplices of Y and let fi (Y ) = |Y (i)|. Let ∆n−1 denote the (n − 1)-dimensional simplex on the vertex set V = [n]. (d−1) For d ≥ 2 let Yd (n, p) denote the probability space of complexes ∆n−1 ⊂ (d) Y ⊂ ∆n−1 with probability measure n Pr(Y ) = pfd (Y ) (1 − p)(d+1)−fd (Y ) .
Let F be an arbitrary fixed field and let Hi (Y ) = Hi (Y ; F) and H i (Y ) = H i (Y ; F) denote the i-th homology and cohomology groups of Y with coefficients in F. Let βi (Y ) = dimF Hi (Y ) = dimF H i (Y ). Kozlov [4] proved the following Theorem 1.1 (Kozlov). For any function ω(n) that tends to infinity ( 1 p = ω(n) n lim Pr [ Y ∈ Yd (n, p) : Hd (Y ) 6= 0 ] = 1 0 p = ω(n)n . n→∞ It is easy to see that if np is bounded away from zero, then the probability that Y ∈ Yd (n, p) contains the boundary of a (d + 1)-simplex does not tend to zero. Thus, the second part of the above statement cannot be improved. Concerning the first part of the statement, as was already observed by Cohen, Costa, Farber and Kappeler for d = 2 (Theorem 6 in [3]), a simple Euler characteristic argument shows that if p = nc where c > d + 1 then a.a.s. Hd (Y ) 6= 0. Our first result is a further improvement on the upper bound in Theorem 1.1. Let gd (x) = (d + 1)(x + 1)e−x + x(1 − e−x )d+1 2
and let cd denote the unique positive solution of the equation gd (x) = d + 1. It is easy to check that gd (d+1) > d+1 and so cd < d+1. A direct calculation yields that cd = d + 1 − Θ( edd ). Theorem 1.2. For a fixed c > cd c lim Pr [Y ∈ Yd (n, ) : Hd (Y ) 6= 0] = 1 . (2) n→∞ n Remark: In the 2-dimensional case, Theorem 1.2 implies that if c > c2 ' 2.783 then Y ∈ Y2 (n, nc ) a.a.s. satisfies H2 (Y ) 6= 0. Simulations indicate that the actual threshold is somewhat lower (around 2.75). We next turn to collapsibility. A (d − 1)−dimensional simplex τ ∈ (d) ∆n−1 (d − 1) is a free face of a complex Y ⊂ ∆n−1 if it is contained in a unique σ ∈ Y (d). Let R(Y ) denote the complex obtained by removing all free (d − 1)-faces of Y together with the d-simplices that contain them. We say that R(Y ) is obtained from Y by a d-collapse step. Let R0 (Y ) = Y and for i ≥ 1 let Ri (Y ) = R(Ri−1 (Y )). We say that Y is d-collapsible if dim R∞ (Y ) < d. Cohen, Costa, Farber and Kappeler [3] proved the following Theorem 1.3 (Cohen, Costa, Farber and Kappeler). If ω(n) → ∞ then 1 ) is a.a.s. 2-collapsible. Y ∈ Y2 (n, ω(n)n Our second result refines Theorem 1.3 and the lower bound in Theorem 1.1 as follows. Let ud (γ, x) = exp(−γ(1 − x)d ) − x . For small positive γ, the only solution of ud (γ, x) = 0 is x = 1. Let γd be the infimum of the set of all nonnegative γ’s for which the equation ud (γ, x) = 0 has a solution x < 1. More explicitly, γd = (dx(1 − x)d−1 )−1 where x satisfies exp(− 1−x ) = x. It is not hard to verify that this yields dx γd = log d + O(log log d). (d−1)
(d)
For ∆n−1 ⊂ Y ⊂ ∆n−1 let s(Y ) denote the number of (d + 1)-simplices in ∆n−1 whose boundary is contained in Y . If c > 0 is fixed and p = nc then a straightforward application of the method of moments (see e.g. Theorem 8.3.1 in [2]) shows that s(Y ) is asymptotically Poisson with parameter c d+2 cd+2 n = . λ = lim E[s] = lim n→∞ n→∞ d + 2 n (d + 2)! 3
The next result asserts that if c < γd and p = nc then s(Y ) > 0 is a.a.s. the only obstruction for d-collapsibility. Let Fn,d denote the family of all (d−1) (d) ∆n−1 ⊂ Y ⊂ ∆n−1 such that s(Y ) = 0. Theorem 1.4. Let c < γd be fixed. Then in the probability space Yd (n, nc ) lim Pr [Y is d − collapsible | Y ∈ Fn,d ] = 1 .
(3)
n→∞
Remark: We have calculated γ2 ' 2.455, and computer simulations suggest that this is indeed the actual threshold for collapsibility for random complexes in Fn,2 . Also, γ3 ' 3.089, γ4 ' 3.508 and γ100 ' 7.555. Clearly, if Y is d-collapsible then Y is homotopy equivalent to a (d − 1)dimensional complex, and in particular Hd (Y ) = 0. Hence for a fixed c < γd and p = nc the following hold: lim Pr [Hd (Y ) = 0 | Y ∈ Fn,d ] = 1
n→∞
and lim Pr [Hd (Y ) = 0] = lim Pr[Fn,d ] = exp(−λ) = exp(−
n→∞
n→∞
cd+2 ). (d + 2)!
The paper is organized as follows. In Section 2 we prove Theorem 1.2. In Section 3 we analyze a random d-tree process that underlies our proof that for c < γd , a random Y ∈ Yd (n, nc ) ∩ Fn,d is a.a.s. d-collapsible. Another main ingredient of the proof is an upper bound on the number of minimal non d-collapsible complexes given in Section 4. In Section 5 we combine these results to derive Theorem 1.4. We conclude in Section 6 with some comments and open problems.
2
The Upper Bound
Let Y ∈ Yd (n, p). Then βi (Y ) = 0 for 0 < i < d − 1 and fi (Y ) = 0 ≤ i ≤ d − 1. The Euler-Poincar´e relation X X (−1)i fi (Y ) = (−1)i βi (Y ) i≥0
i≥0
4
n i+1
for
therefore implies n−1 βd (Y ) = fd (Y ) − + βd−1 (Y ). (4) d The inequality βd (Y ) ≥ fd (Y ) − n−1 already implies that if c > d + 1 d then a.a.s. βd (Y ) 6= 0. As mentioned above, this was observed in the 2dimensional case by Cohen, Costa, Farber and Kappeler [3]. The idea of the proof of Theorem 1.2 is to improve this estimate by providing a non-trivial lower bound on E[βd−1 ]. For τ ∈ ∆n−1 (d − 1) let degY (τ ) = |{σ ∈ Y (d) : τ ⊂ σ}| and let Aτ = {Y ∈ Yd (n, p) : degY (τ ) = 0} . For σ ∈ Y (d) let Lσ be the subcomplex of σ (d−1) given by Lσ = σ (d−2) ∪ {τ ∈ σ (d−1) : degY (τ ) > 1} . Let Pn,d denote the family of all pairs (σ, L), such that σ ∈ ∆n−1 (d) and σ (d−2) ⊂ L ⊂ σ (d−1) . For (σ, L) ∈ Pn,d let Bσ,L = {Y ∈ Yd (n, p) : σ ∈ Y , Lσ = L} . The space of i-cocycles of a complex K is as usual denoted by Z i (K). The space of relative i-cocycles of a pair K 0 ⊂ K is denoted by Z i (K, K 0 ) and will be identified with the subspace of i-cocycles of K that vanish on K 0 . Let z i (K) = dim Z i (K) and z i (K, K 0 ) = dim Z i (K, K 0 ). For a (d − 1)-simplex τ = [v0 , . . . , vd−1 ], let 1τ be the indicator (d − 1)-cochain of τ (i.e. 1τ (η) = sgn(π) if η = [vπ(0) , . . . , vπ(d−1) ] and is zero otherwise). If τ ∈ ∆n−1 (d − 1) then Z d−1 (τ ) is the 1-dimensional space spanned by 1τ . If (σ, L) ∈ Pn,d and fd−1 (L) = j, then z d−1 (σ, L) = d − j. indeed, suppose σ = [v0 , . . . , vd ] and for 0 ≤ i ≤ d let τi = [v0 , . . . , vbi , . . . , vd ]. If L(d − 1) = {τi }di=d−j+1 then d−1 {1τ0 − 1τi }d−j (σ, L) . i=1 forms a basis of Z Claim 2.1. For any Y ∈ Yd (n, p) M Z d−1 (Y ) ⊃ Z d−1 (τ ) ⊕ {τ ∈∆n−1 (d−1) : Y ∈Aτ }
M {(σ,L)∈Pn,d : Y ∈Bσ,L }
5
Z d−1 (σ, L) .
Proof: The containment is clear. To show that the right hand side is a direct sum note that nontrivial cocycles in different summands must have disjoint supports and are therefore linearly independent. Let a(Y ) = |{τ ∈ ∆n−1 (d − 1) : degY (τ ) = 0}| and for 0 ≤ j ≤ d let αj (Y ) = |{(σ, L) ∈ Pn,d : Y ∈ Bσ,L , fd−1 (L) = j}|. Note that αj (Y ) is the number of d-faces of Y that contain exactly d + 1 − j (d − 1)-faces of degree 1. By Claim 2.1 z
d−1
def
(Y ) ≥ u(Y ) = a(Y ) +
d X
αj (Y )(d − j) .
(5)
j=0 n−1 d−1
, it follows from (4) and (5) n def βd (Y ) ≥ v(Y ) = fd (Y ) + u(Y ) − . (6) d Theorem 1.2 will thus follow from As βd−1 (Y ) = dim H d−1 (Y ) = z d−1 (Y ) − that
Theorem 2.2. Let c > cd and let p = nc . Then lim Pr [ Y ∈ Yd (n, p) : v(Y ) ≤ 0] = 0 .
n→∞
Proof: First note that
n c E[fd ] = p= nd − O(nd−1 ) , d+1 (d + 1)! n e−c d E[a] = (1 − p)n−d = n − O(nd−1 ) , d d! and for 0 ≤ j ≤ d
n d+1 E[αj ] = p(1 − p)(n−d−1)(d+1−j) (1 − (1 − p)n−d−1 )j d+1 j nd c d + 1 −c(d+1−j) = e (1 − e−c )j − O(nd−1 ) . (d + 1)! j 6
Therefore E[u] = E[a] +
d X
E[αj ](d − j)
j=0
d nd e−c nd c X d + 1 −c(d+1−j) = + e (1 − e−c )j (d − j) − O(nd−1 ) d! (d + 1)! j=0 j =
nd ((1 + c)(d + 1)e−c − c(1 − (1 − e−c )d+1 )) − O(nd−1 ). (d + 1)!
It follows that n E[v] = E[fd ] + E[u] − d d n (c + (1 + c)(d + 1)e−c − c(1 − (1 − e−c )d+1 ) − (d + 1)) − O(nd−1 ) = (d + 1)! nd = (gd (c) − d − 1) − O(nd−1 ) . (d + 1)! Since c > cd it follows that for sufficiently large n E[v] ≥ nd
(7)
where > 0 depends only on c and d. To show that v is a.a.s. positive we use the following consequence of Azuma’s inequality due to McDiarmid [6]. Theorem 2.3. Suppose f : {0, 1}m → R satisfies |f (x) − f (x0 )| ≤ T if x and x0 differ in at most one coordinate. Let ξ1 , . . . , ξm be independent 0, 1 valued random variables and let F = f (ξ1 , . . . , ξm ). Then for all λ > 0 Pr[F ≤ E[F ] − λ] ≤ exp(−
2λ2 ). T 2m
(8)
n Let m = d+1 and let σ1 , . . . , σm be an arbitrary ordering of the dsimplices of ∆n−1 . Identify Y ∈ Yd (n, p) with its indicator vector (ξ1 , . . . , ξm ) where ξi (Y ) = 1 if σi ∈ Y and ξi (Y ) = 0 otherwise. Note that if Y and Y 0 differ in at most one d-simplex then |a(Y ) − a(Y 0 )| ≤ d + 1 and |αj (Y ) − αj (Y 0 )| ≤ d + 1 for all 0 ≤ j ≤ d. It follows that |v(Y ) − v(Y 0 )| ≤ T = 2d3 .
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Applying McDiarmid’s inequality (8) with F = v and λ = E[v] it follows that 2E[v]2 Pr[v ≤ 0] ≤ exp(− 3 2 ) ≤ exp(−C2 nd−1 ) (2d ) m for some C2 = C2 (c, d) > 0.
Remark: The approach used in the proof of Theorem 1.2 can be extended as d−1 follows. For a fixed `, let Z(`) (Y ) ⊂ Z d−1 (Y ) denote the subspace spanned by (d − 1)-cocycles φ ∈ Z d−1 (Y ) such that |supp(φ)| ≤ `. Let d−1 E[dim Z(`) (Y )] θd,` (x) = lim n n→∞
d
where the expectation is taken in the probability space Yd (n, nx ). For example, it was shown in the proof of Theorem 1.2 that θd,1 (x) = e−x and θd,2 (x) = (1 + x)e−x −
x (1 − (1 − e−x )d+1 ). d+1
Let x = cd,` denote the unique positive root of the equation x + (d + 1)θd,` (x) = d + 1. The following fact is implicit in the proof of Theorem 1.2. Proposition 2.4. For any fixed c > cd,` c lim Pr [Y ∈ Yd (n, ) : Hd (Y ) 6= 0] = 1 . n→∞ n
(9)
Let c˜d = lim`→∞ cd,` . It seems likely that c˜nd is the exact threshold for the vanishing of H d (Y ). This is indeed true in the graphical case d = 1. Proposition 2.5. c˜1 = 1.
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Proof: For a subtree K = (VK , EK ) on the vertex set VK ⊂ [n] let AK denote all graphs G ∈ G(n, p) that contain K as an induced subgraph and contain no edges in the cut (VK , VK ). The space of 0-cocycles Z 0 (K) is 1-dimensional and is spanned by the indicator function of VK . As in Claim 2.1 it is clear that for G ∈ G(n, p) and a fixed ` M Z 0 (G) ⊃ Z 0 (K). {K:|VK |≤` and G∈AK }
Hence for p =
x n
0 E[dim Z(`) (G)] =
X {Pr[AK ] : K is a tree on ≤ ` vertices} =
` X k−1 n k−2 x k−1 x k ( ) (1 − )k(n−k)+( 2 ) ∼ n n k k=1
n
` X k k−2 k=1
P∞
kk−2
Let S(z) = k=1 ber of trees. Then
k!
k!
`
x
k−1 −xk
e
z k be the exponential generating function for the num-
lim θ1,` (x) = lim
`→∞
n X k k−2 = (xe−x )k . x k=1 k!
0 (G)] E[dim Z(`)
n
`,n→∞
=
S(xe−x ) . x
Therefore c˜1 = lim`→∞ c1,` is the solution of the equation x+
2S(xe−x ) = 2. x
(10)
P∞ kk−1 k Let R(z) = k=1 k! z be the exponential generating function for the number of rooted trees. It is classically known (see e.g. [7]) that R(z) = z exp(R(z)), and that S(z) = R(z) − 21 R(z)2 . It follows that R(e−1 ) = 1 and S(e−1 ) = 21 . Hence c˜1 = 1 is the unique solution of (10).
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3
The Random d-Tree Process
A simplicial complex T on the vertex set V with |V | = ` ≥ d is a d-tree if there exists an ordering V = {v1 , . . . , v` } such that lk(T [v1 , . . . , vi ], vi ) is a (d − 1)-dimensional simplex for all d + 1 ≤ i ≤ `. Let GT denote the graph with vertex set T (d − 1), whose edges are the pairs {τ1 , τ2 } such that τ1 ∪ τ2 ∈ T (d). Let distT (τ1 , τ2 ) denote the distance between τ1 and τ2 in the graph GT . A rooted d-tree is a pair (T, τ ) where T is a d-tree and τ is some (d−1)-face of T . Let τ be a fixed (d − 1)-simplex. Given k ≥ 0 and γ > 0 we describe a random process that gives rise to a probability space Td (k, λ) of all d-trees T rooted at τ such that distT (τ, τ 0 ) ≤ k for all τ 0 ∈ T . The definition of Td (k, λ) proceeds by induction on k. Td (0, γ) is the (d − 1)-simplex τ . Let k ≥ 0. A d-tree in Td (k + 1, γ) is generated as follows: First generate a T ∈ Td (k, λ) and let U denote all τ 0 ∈ T (d − 1) such that distT (τ, τ 0 ) = k. Then, independently for each τ 0 ∈ U, pick J = Jτ 0 new vertices z1 , . . . , zJ where J is Poisson distributed with parameter γ, and add the d-simplices z1 τ 0 , . . . , zJ τ 0 to T . We next define the operation of pruning of a rooted d-tree (T, τ ). Let {τ1 , . . . , τ` } be the set of all free (d−1)-faces of T that are distinct from τ , and let σi be the unique d-simplex of T that contains τi . The d-tree T 0 obtained from T by removing the simplices τ1 , σ1 , . . . , τ` , σ` is called the pruning of T . Clearly, any T ∈ Td (k +1, γ) collapses to its root τ after at most k +1 pruning steps. Denote by Cd (k + 1, γ) the event that T ∈ Td (k + 1, γ) collapses to τ after at most k pruning steps, and let ρd (k, γ) = Pr[Cd (k + 1, γ)]. Clearly, ρd (0, γ) is the probability that T ∈ Td (1, γ) consists only of τ , hence ρd (0, γ) = e−γ .
(11)
Let σ1 , . . . , σj denote the d-simplices of T ∈ Td (k + 1, γ) that contain τ and for each 1 ≤ i ≤ j let ηi1 , . . . , ηid be the (d − 1)-faces of σi that are different from τ . Let Ti` ∈ Td (k, λ) denote the subtree of T that grows out of ηi` . Clearly, T collapses to τ after at most k pruning steps iff for each 1 ≤ i ≤ j, at least one of the d-trees Ti` collapses to its root ηi` in at most k − 1 steps.
10
We therefore obtain the following recursion: ρd (k, γ) = =
∞ X j=0 ∞ X j=0
Pr [J = j](1 − (1 − ρd (k − 1, γ))d )j γ j −γ e (1 − (1 − ρd (k − 1, γ))d )j j!
(12)
= exp(−γ(1 − ρd (k − 1, γ))d ) . Equations (11) and (12) imply that the sequence {ρd (k, γ)}k is non-decreasing and converges to ρd (γ) ∈ (0, 1], where ρd (γ) is the smallest positive solution of the equation ud (γ, x) = exp(−γ(1 − x)d ) − x = 0. (13) If γ ≥ 0 is small, then ρd (γ) = 1. Let γd denote the infimum of the set of nonnegative γ’s for which ρd (γ) < 1. The pair (γ, x) = (γd , ρd (γd )) satisfies both d (γd , ρd (γd )) = 0. A straightforward computation ud (γd , ρd (γd )) = 0 and ∂u ∂x shows that γd = (dx(1 − x)d−1 )−1 where x = ρd (γd ) is the unique solution of exp(− 1−x ) = x. dx
4
The Number of Non-d-Collapsible Complexes
When we discuss d-collapsibility, we only care about the inclusion relation between d-faces and (d − 1)-faces. Therefore, in the present section we can and will simplify matters and consider only the complex that is induced from our (random) choice of d-faces. Namely, for every i ≤ d, a given i-dimensional face belongs to the complex iff it is contained in some of the chosen d-faces. A complex is a core if every (d − 1)-dimensional face belongs to at least two simplices, so that not even a single collapse step is possible. A core complex is called a minimal core complex if none of its proper subcomplexes is a core. The main goal of this section is to show that with almost certainty there are just two types of minimal core subcomplexes that a sparse random complex can have. It can either be the boundary of a (d + 1)-simplex, ∂∆d+1 , or it must be very large. Obviously this implies that there are no small noncollapsible subcomplexes which do not contain the boundary of a (d + 1)simplex.
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Theorem 4.1. For every c > 0 there exists a constant δ = δ(c) > 0 such that a.a.s. every minimal core subcomplex K of Y ∈ Yd (n, nc ) with fd (K) ≤ δnd , must contain the boundary of a (d + 1)-simplex. Henceforth we use the convention that faces refer to arbitrary dimensions, but unless otherwise specified, the word simplex is reserved to mean a d-face. Our proof uses the first moment method. In the main step of the proof we obtain an upper bound on Cd (n, m), the number of all minimal core ddimensional complexes on vertex set [n] = {1, 2, . . . , n}, which contain m simplices. Two simplices are considered adjacent if their intersection is a (d−1)- face. · If A ∪ B is a splitting of a minimal core complex, then there is a simplex in A and one in B that are adjacent, otherwise the corresponding subcomplexes are cores as well. Therefore K can be constructed by successively adding a simplex that is adjacent to an already existing simplex. This consideration easily yields an upper bound of nd+m on Cd (n, m). The point is that if m = δnd for δ > 0 small enough, we get an exponentially smaller (in m) upper bound and this is crucial for our analysis. Lemma 1. Let m = δnd and δ > 0 small enough. Then 1 nd−1 (14) nd nm (2d+1 d3 δ d4 )m . Cd (n, m) ≤ d−1 2 (d m) d 1 d(d+1)δ d Proof. Let b = . A (d − 2)-face is considered heavy or light de2 pending on whether it is covered by at least bn (d − 1)-faces or less. The sets of heavy and light (d − 2)-faces are denoted by Hd−2 and Ld−2 repectively. We claim that |Hd−2 | ≤ bd−1 nd−1 . To see this note that each simplex contains exactly d + 1 (d − 1)-faces, but the complex is a core, so that each (d − 1)-face is covered at least twice. Consequently, our complex has at most m(d+1) (d − 1)-faces. Likewise, each (d − 1)-face contains d (d − 2)-faces. Each 2 heavy (d − 2)-face is covered at least bn times and the claim follows by the following calculation:
|Hd−2 | ≤
d(d + 1)m d(d + 1)δnd d(d + 1)δnd−1 = = = bd−1 nd−1 . 2bn 2bn 2b
We extend the heavy/light dichotomy to lower dimensions as well. For each 0 ≤ i ≤ d − 3, an i-face is considered heavy if it covered by at least b · n 12
heavy (i + 1)-faces. Otherwise it is light. The sets of heavy/light i-faces are denoted by Hi resp. Li . By counting inclusion relations between heavy faces i+1 | which yields of consecutive dimensions it is easily seen that |Hi | ≤ (i+2)|H bn |Hi | ≤
(d − 1)! i+1 i+1 b n . (i + 1)!
The set of i-dimensional heavy (resp. light) faces contained in a given face σ is denoted by Hiσ be (resp. Lσi ). The bulk of the proof considers a sequence of complexes C1 , . . . , Cm = C, where the complex Ci is obtained from Ci−1 by adding a single simplex. A (d−1)-face σ of Ci can be saturated or unsaturated. This depends on whether or not every simplex in Cm that contains σ already belongs to Ci . Prior to defining the complexes Ci , we specify the set of heavy (d − 2)-faces in one of n at most (d−1) possible ways. Note that this choice uniquely determines bd−1 nd−1
the sets of heavy faces for every dimension 0 ≤ i ≤ d − 3. We start off n with the complex C1 , which has exactly one simplex. Clearly there are d+1 possible choices for C1 . We move from Ci−1 to Ci by adding a single simplex ti , which covers a chosen unsaturated (d − 1)-face σi−1 of Ci−1 . Our choices are subject to the condition that every heavy (d − 2)-face in Cm is one of the heavy (d − 2)-faces chosen prior to the process. In other words, we must never make choices that create any additional heavy faces in addition to those derived from our preliminary choice. Our goal is to bound the number of choices for this process. The crux of the argument is a rule for selecting the chosen face. Associated with every face is a vector counting the number of its heavy vertices, its heavy edges, its heavy 2-faces etc. The chosen face is always lexicographically minimal w.r.t. this vector, breaking ties arbitrarily. A (d − 1)-face all of whose subfaces are light is called primary. In each step j we expand a (d − 1)-face σ to a simplex σ ∪ y. Such a step is called a saving step if either: 1. The vertex y is heavy. 2. There exists a light (d − 2)- subface τ ⊂ σ such that τ ∪ y is contained in a simplex in Cj−1 . 3. There exists a light subface τ ⊂ σ such that the face τ ∪ y is heavy.
13
Note that the number of choices of y in the first case is ≤ |H0 | ≤ (d − 1)! · b · n. In the second case the number of choices for y is at most dbn. In the third case there are d − 2 possibilities for the dimension of the d light face and for each such dimension i there are at most i+1 bn choices for y. In all cases the number of choices for y is at most ≤ dd bn. A step that is not saving is considered wasteful. For wasteful steps we bound the number of choices for y by n. The idea of the proof is that every such a process which produces a minimal core complex must include many saving steps. More specifically, we want to show: Claim 4.2. For every d3 wasteful steps, at least one saving step is carried out. Proof. The proof of this claim consists of two steps. We show that there is no sequence of d(d − 1) consecutive wasteful steps, without the creation of an unsaturated primary face. Also, the creation of d + 1 primary faces necessarily involves a saving step. If u is a vertex in a (d − 1)-face σ, let riσ (u) be the number of heavy i-faces in σ that contain u. Also, Viσ denotes the set of vertices v in σ that are included only in light i-subfaces of σ. Proposition 4.3. Let σ and σ 0 be two consecutively chosen faces where σ is non-primary and the extension step on σ is wasteful. Then σ 0 precedes σ in 0 the order of faces and |Viσ | ≥ |Viσ | + 1 where i is the smallest dimension for which |Hiσ | > 0. Proof. Since the extension step on σ is wasteful (and, in particular, not a saving step of type (iii)) and since all j-subfaces of σ are light for j < i, every j-face in σ ∪ y is light. Moreover, every i-subface of σ ∪ y that contains y is light as well. We claim that σ 0 = σ \{u}∪{y} where the vertex u of σ maximizes riσ (v). (Since |Hiσ | > 0, there are vertices v in σ for which riσ (v) > 0). Notice that σ has d−1 −riσ (u) more light i-subfaces than does τ u := σ\u. i u Namely, |Lτi | = |Lσi | − d−1 + riσ (u). i Combining the fact that every i-subface of σ ∪ y that contains y is light τyu u u u we see that in τy := τ ∪ y, |Li | = |Lτi | + d−1 = |Lσi | + riσ (u). But since i σ u u ri (u) > 0, τy precedes σ. In this case τy must be a new face that does not 14
belong to the previous complex, or else it would have been preferred over σ. Being a new face, it is necessarily unsaturated. Since u maximizes riσ (u) over all vertices in σ, it follows that τyu precedes all other faces created in the expansion. Furthermore, no other face precedes σ or else it would be chosen 0 rather than σ. Thus σ 0 = τuy , as claimed. Notice that y ∈ Viσ and also 0 Viσ ⊆ Viσ (Note that every i-dimensional subfaces of σ 0 that is not contained 0 in σ is light since it contains y). Thus |Viσ | ≥ |Viσ | + 1. Consider a chosen non-primary face σ and let i be the smallest dimension for which |Hiσ | > 0. The previous claim implies that after at most d consecutive wasteful steps the chosen face, θ precedes σ and |Viθ | = d. Then |Hjθ | = 0 for all j ≤ i (in particular |Hiθ | = 0). By repeating this argument d − 1 times we conclude that following every series of d(d − 1) consecutive wasteful steps, a primary face must be chosen: After at most d consecutive wasteful steps the chosen face can have no heavy vertices. At the end of the next d consecutive wasteful steps, the chosen face has no heavy vertices nor heavy edges. Repeating this argument (d − 1) times necessarily leads us to a chosen primary face. Proposition 4.4. Only saving steps can decrease the number of unsaturated primary faces. Proof. Let σ = a1 , a2 . . . , ad be a primary face and let y be the vertex that expands it. Denote the (d − 1)-face {a1 , a2 . . . , ai−1 , ai+1 , . . . , ad } ∪ {y} by σ i . Since this is not a saving step of type (i), y is light. It is also not of type i (iii) and so |Hkσ | = 0 for every i = 1, . . . , d and k = 1, . . . , d − 2, so that faces σ i are primary. However this is not a type (ii) saving step, so all the (d − 1)-faces σ i must be new. Thus the number of unsaturated primary faces has increased by at least d − 1. The proof of Claim 4.2 is now complete, since at each step at most d + 1 faces get covered. We can turn now to bound the number of minimal core m-simplex complexes Cm . As mentioned, we first specify the heavy (d − 2)-faces of Cm by n d−1 d−1 specifying a set of b n out of the total of d−1 (d − 2)-faces. Then we select the first simplex C1 and mark all its (d − 1)-faces as unsaturated. In order to choose the i-th step we first decide whether it is a saving or wasteful step, and if it is a saving step, what type it has. There is a total of d + 1 possible kinds of extensions of the current (d − 1)-face: A saving step of type 15
(i), (ii), or one of the d − 2 choices of type (iii) (according to dimension), or a wasteful step. In a saving step the expanding vertex can be chosen in at most dd bn ways. The number of possible extension clearly never exceeds n and it is this trivial upper bound that we use for wasteful steps. Finally we update the labels on the (d − 1)-faces of a new simplex. We need to decide which of the unsaturated (d − 1)-faces that are already covered by at least two simplices change their status to saturated. There are at most 2d+1 possibilities of such an update. As we saw, at least dm3 of the steps in such process are saving steps. Consequently we get the following upper bound on Cd (n, m), the number of minimal core n-vertex d-dimensional complexes with m simplices. (In reading the expression below, note that the terms therein correspond in a one-to-one manner to the ingredients that were just listed).
n d−1 nd+1 (d bd−1 nd−1
m
+ 1)m−1 nm−1 (dd b) d3 (2d+1 )m nd−1 d m d+1 d12 d13 m n n ((d + 1)2 d b ) ≤ d−1 (d2 δ) d nd−1 1 nd−1 nd−1 d m d+1 2 2 d14 m ≤ n n (2 d (d δ) ) ≤ nd nm (2d+1 d3 δ d4 )m . d−1 d−1 2 2 (d m) d (d m) d
Proof of Theorem 4.1: We show the assertion with δ = δ(c) = 4 (2d+2 d3 c)−d . Indeed, consider a complex drawn from Yd (n, p). Let Xm = Xm (n, p) count the number of minimal core subcomplexes with m simplices and which are not copies of ∂∆d+1 . Our argument splits according to whether m is small or large, the dividing line being m = m1 = (d3 log n)d . The theorem speaks only about the range m ≤ m2 = δ(c)nd . By Lemma 1,
16
m2 X
EXm ≤
m
Cd (n, m)p ≤
m=m1
m=m1
≤ nd
m2 X
m2 X
d−1 2 m) d
(nd−1 )(d
1
nd 2d+1 d3 δ d4 c
m
m=m1
d−1 2 m2 1 (d m) d d−1 X 2 d m (d m) d nd−1 nd−1 2− d2 2−m ≤ nd
m2 X m=m1
m=m1
m2 X 1 ≤n n m=m
d−1 (d2 m) d
d
= o(1)
1
It follows that with almost certainty no cores with m simplices occur where m2 ≥ m ≥ m1 . We next consider the range d + 3 ≤ m ≤ m1 . Note d+3 that a minimal core complex with m ≥ d + 3 simplices has at most d+4 m vertices. Let ∆(u) denote the number of simplices that contain the vertex u. Clearly, if ∆(u) > 0 then ∆(u) ≥ d + 1 (Consider a simplex σ that contains u. Every face of the form σ \ w with u 6= w ∈ σ is covered by a simplex other than σ). It is not hard to verify that if some simplex σ contains two distinct vertices with ∆(u) = ∆(w) = d + 1 then the complex contains ∂∆d+1 contrary to the minimality assumption. Let t be the number of vertices u m . with ∆(u) = d + 1. No simplex contains two such vertices, so that t ≤ d+1 Counting vertices in the complex according to the value of ∆ we get (d + 1)t + (d + 2)(v − t) ≤ (d + 1)m where v is the total number of vertices. The conclusion follows. The expected number of minimal core subcomplexes of Yd (n, p) that contain d + 3 ≤ m ≤ m1 simplices satisfies h m1 m1 m1 h im X X X d+3 d + 3 d+1 im n m d+1 d+4 m p EXm ≤ m p ≤ n d+3 m d+4 d+4 m=d+3 m=d+3 m=d+3 ≤
m1 X m=d+3
cd+4
m log2d(d+1)(d+4) n d+4 = o(1). n
Consequently, a.a.s. Yd (n, p) contains no minimal core subcomplexes of m 4 simplices with d + 3 ≤ m ≤ (2d+2 d3 c)−d nd . 17
5
The Threshold for d-Collapsibility (d)
For a complex Y ⊂ ∆n−1 and a fixed τ ∈ ∆n−1 (d − 1), define a sequence of complexes {Si (Y )}i≥0 as follows. S0 (Y ) = τ and for i ≥ 1 let Si (Y ) be the union of Si−1 (Y ) and the complex generated by all the d-simplices of Y that contain some η ∈ Si−1 (Y )(d − 1). Let Td denote the family of all d-trees. Consider the events Ak , D ⊂ Yd (n, p) given by Ak = {Sk (Y ) ∈ Td } and D = {degY (η) ≤ log n for all η ∈ ∆n−1 (d − 1)}. Claim 5.1. Let k and c > 0 be fixed and p = nc . Then Pr[Ak+1 ∩ D] = 1 − o(1). Proof: Fix η ∈ ∆n−1 (d − 1). The random variable degY (η) has a binomial distribution B(n − d, nc ), hence by the large deviations estimate Pr[degY (η) > log n] < n−Ω(log log n) . Therefore Pr[D] = 1 − o(1). If Y ∈ D then f0 (Sk+1 (Y )) = O(logk+1 n) and fd−1 (Sk (Y )) = O(logk n). Note that Sk+1 (Y ) is a d-tree iff in its generation process, we never add a simplex of the form ηv such that both η ∈ ∆n−1 (d−1) and v ∈ ∆n−1 (0) already exist in the complex. Since the number of such pairs is at most f0 (Sk+1 (Y ))fd−1 (Sk (Y )) it follows that Pr[Ak+1 ∩ D] ≥ (1 − (1 −
c f0 (Sk+1 (Y ))fd−1 (Sk (Y )) ) ≥ n
c O(log2k+1 n) ) = 1 − o(1) . n
(d)
For Y ⊂ ∆n−1 let r(Y ) = fd (R∞ (Y )) be the number of d-simplices remaining in Y after performing all possible d-collapsing steps. For τ ∈ ∆n−1 (d − 1) let Γ(τ ) = {σ ∈ ∆n−1 (d) : σ ⊃ τ }. Claim 5.2. Let 0 < c < γd be fixed and p = nc . Then for any fixed τ ∈ ∆n−1 (d): Pr[R∞ (Y ) ∩ Γ(τ ) 6= ∅] = o(1) . (15) 18
Proof: Let δ > 0. Since c < γd lim ρd (k, c) = ρd (c) = 1.
k→∞
Choose a fixed k such that δ ρd (k, c) > 1 − . 3 Claim 5.1 implies that if n is sufficiently large then δ Pr[Ak+1 ∩ D] ≥ 1 − . 3 Next note that if Y ∈ Ak+1 then Sk+1 = Sk+1 (Y ) can be generated by the following inductively defined random process: S0 = τ . Let 0 ≤ i ≤ k. First generate T = Si and let U denote all τ 0 ∈ T (d − 1) such that distT (τ, τ 0 ) = i. Then, according to (say) the lexicographic order on U, for each τ 0 ∈ U pick J new vertices z1 , . . . , zJ according to the binomial distribution B(n − n0 , nc ), where n0 is the number of vertices that appeared up to that point, and add the d-simplices z1 τ 0 , . . . , zJ τ 0 to T . Note that the process described above is identical to the d-tree process of Section 3, except for the use of the binomial distribution B(n − n0 , nc ) instead of the Poisson distribution P o(c). Now if Y ∈ Ak+1 ∩ D then n0 = O(logk+1 n) at all stages of this process. It follows that if n is sufficiently large then the total variation distance between the distributions Sk+1 (Y ) and Td (k+1, c) is less then 3δ . Denote by Cd0 (k+1, c) the event that Sk+1 (Y ) is in Ak+1 and collapses to τ in at most k pruning steps. The crucial observation now is that if Y ∈ Cd0 (k+1, c) then R∞ (Y )∩Γ(τ ) = ∅. It follows that Pr[R∞ (Y ) ∩ Γ(τ ) 6= ∅] ≤ (1 − Pr[Ak+1 ∩ D]) + Pr[Y 6∈ Cd0 (k + 1, c)] ≤ (1 − Pr[Ak+1 ∩ D]) + dT V (Sk+1 (Y ), Td (k + 1, c)) + (1 − Pr[Cd (k + 1, c)]) ≤ δ δ δ + + = δ. 3 3 3 Let G(Y ) = {τ ∈ ∆n−1 (d − 1) : R∞ (Y ) ∩ Γ(τ ) 6= ∅} and let g(Y ) = |G(Y )|. For a family G ⊂ ∆n−1 (d − 1) let w(G) denote the set of all d-simplices σ ∈ ∆n−1 (d) all of whose (d − 1)-faces are contained in G. Using Claim 5.2 we establish the following 19
Theorem 5.3. Let δ > 0 and 0 < c < γd be fixed and let p = nc . Then Pr[fd (R∞ (Y )) > δnd ] = o(1). Proof: Let 0 < = (d, c, δ) < 1 be a constant whose value will be fixed later. Clearly Pr[fd (R∞ (Y )) > δnd ] ≤ Pr[g(Y ) > δnd ] + Pr[g(Y ) ≤ δnd
& fd (R∞ (Y )) > δnd ].
To bound the first summand note that E[g] = o(nd ) by Claim 5.2. Hence by Markov’s inequality Pr[g(Y ) > δnd ] ≤ (δnd )−1 E[g] = o(1). Next note that Pr[g(Y ) ≤ δnd X
& fd (R∞ (Y )) > δnd ] ≤ Pr[|w(G) ∩ Y (d)| > δnd ].
{G⊂∆n−1 (d−1):|G|=δnd }
Fix a G ⊂ ∆n−1 (d−1) such that |G| = δnd . By the Kruskal-Katona theorem there exists a C1 = C1 (d, δ) such that N = |w(G)| ≤ C1
d+1 d
nd+1 .
Applying the large deviation estimate for the binomial distribution B(N, nc ) 1 and writing C2 = ecC we obtain δ Pr[|w(G) ∩ Y (d)| > δnd ] ≤ (C2 On the other hand
n d
δnd Choosing such that (
≤(
d+1 d
d
)δn .
e δnd ) . δ
d+1 e ) C2 d < e−1 δ
it follows that Pr[g(Y ) ≤ δnd
& fd (R∞ (Y )) > δnd ] ≤ exp(−δnd ). 20
Proof of Theorem 1.4: Let c < γd and p = nc . By Theorem 4.1 there exists a δ > 0 such that a.a.s. any non-d-collapsible subcomplex K of Y ∈ Yd (n, nc ) such that fd (K) ≤ δnd contains the boundary of a (d + 1)-simplex. It follows that Pr [Y non d − collapsible | Y ∈ Fn,d ] = Pr [fd (R∞ (Y )) > 0 | Y ∈ Fn,d ] ≤ Pr [fd (R∞ (Y )) > δnd ]·Pr [ Y ∈ Fn,d ]−1 +Pr [0 < fd (R∞ (Y )) ≤ δnd | Y ∈ Fn,d ]. The first summand is o(1) by Theorem 5.3, and the second summand is o(1) by Theorem 4.1.
6
Concluding Remarks
Let us remark that one may show a random process statement slightly stronger than Theorem 4.1 (see [5], where a similar result is shown for the k-core of random graphs). More specifically, let us define the d-dimensional (n) random process Yd = {Yd (n, M )}Md+1 =0 as the Markov chain whose stages are simplicial complexes, which starts with the full (d − 1)-dimensional skeleton of ∆n−1 and no d-simplices, and in each stage Yd (n, M + 1) is obtained from Yd (n, M ) by adding to it one d-simplex chosen uniformly at random from all the d-simplices which do not belong to Yd (n, M ). The core of a complex Y is the maximal core subcomplex of Y . Then, the following holds. Theorem 6.1. There exists a constant α = α(d) > 0 such that for almost ev(n) ery d-dimensional random process Yd = {Yd (n, M )}Md+1 =0 there exists a stage ˆ ˆ ˆ M = M (Yd ) such that the core of Yd (n, M ) is of the size O(1) and consists ˆ + 1) contains at of boundaries of (d + 1)-simplices, while the core of Yd (n, M d least αn d-simplices. Many questions remain open. The most obvious ones are • What is the threshold for d-collapsibility of random simplicial complexes in Fn,d ? We conjecture that it is indeed p = γd /n. 21
• Find the exact threshold for the nonvanishing of Hd (Y ). The first two authors (see [1]) have recently improved the upper bound given in Theorem 1.2 and they conjecture that their new bound is in fact sharp. This in particular would imply that the threshold does not depend on the underlying field. • Although this question is implicitly included in the above two questions, it is of substantial interest in its own right: Can you show that the two thresholds (for d-collapsibility and for the vanishing of the top homology) are distinct? We conjecture that the two thresholds are, in fact, quite different. In particular, although d-collapsibility is a sufficient condition for the vanishing of Hd , there is only a vanishingly small probability that a random simplicial complex with trivial top homology is d-collapsible.
References [1] L. Aronshtam and N. Linial, When does the top homology of a random simplicial complex vanish?, arXiv:1203.3312. [2] N. Alon and J. Spencer, The Probabilistic Method, 2nd Edition, WileyIntescience, 2000. [3] D. Cohen, A. Costa, M. Farber and T. Kappeler, Topology of Random 2-Complexes, Discrete Comput. Geom. 47(2012) 117–149. [4] D. Kozlov, The threshold function for vanishing of the top homology group of random d-complexes, Proc. Amer. Math. Soc. 138(2010) 4517– 4527. [5] T. Luczak, Size and connectivity of the k-core of a random graph, Discrete Math. 91(1991) 61–68. [6] C. McDiarmid, On the Method of Bounded Differences, Surveys in combinatorics 1989, 148–188, London Math. Soc. Lecture Note Ser., 141, Cambridge Univ. Press, Cambridge, 1989. [7] R. Stanley, Enumerative Combinatorics Vol. I, Cambridge Unversity Press, Cambridge, 1997.
22
N. Linial†
T. Luczak‡
R. Meshulam§
November 2, 2012
Abstract Let ∆n−1 denote the (n − 1)-dimensional simplex. Let Y be a random d-dimensional subcomplex of ∆n−1 obtained by starting with the full (d − 1)-dimensional skeleton of ∆n−1 and then adding each d-simplex independently with probability p = nc . We compute an explicit constant γd , with γ2 ' 2.45, γ3 ' 3.5 and γd = Θ(log d) as d → ∞, so that for c < γd such a random simplicial complex either collapses to a (d − 1)-dimensional subcomplex or it contains ∂∆d+1 , the boundary of a (d + 1)-dimensional simplex. We conjecture this bound to be sharp. In addition we show that there exists a constant γd < cd < d + 1 such that for any c > cd and a fixed field F, asymptotically almost surely Hd (Y ; F) 6= 0.
1
Introduction
Let G(n, p) denote the probability space of graphs on the vertex set [n] = {1, . . . , n} with independent edge probabilities p. It is well known (see e.g. ∗
Department of Computer Science, Hebrew University, Jerusalem 91904, Israel. e-mail: [email protected] . † Department of Computer Science, Hebrew University, Jerusalem 91904, Israel. e-mail: [email protected] . Supported by ISF and BSF grants. ‡ Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Pozna´ n, Poland. e-mail: [email protected] . Supported by the Foundation for Polish Science. § Department of Mathematics, Technion, Haifa 32000, Israel. e-mail: [email protected] . Supported by ISF and ESF grants.
1
[2]) that if c ≥ 1 then a graph G ∈ G(n, nc ) a.a.s. contains a cycle, while for a constant c < 1 √ c c2 c (1) lim Pr [ G ∈ G(n, ) : G acyclic ] = 1 − c · exp( + ). n→∞ n 2 4 In this paper we consider the analogous question for d-dimensional random complexes. There are two natural extensions to the notion of an acyclic graph. Namely, the vanishing of the d-th homology, and collapsibility to a (d − 1)-dimensional subcomplex. These are the two questions we consider here. We provide an upper bound on the threshold for the vanishing of the d-th homology and a lower bound (which we believe to be tight) for the threshold for collapsibility. For a simplicial complex Y , let Y (i) denote the i-dimensional skeleton of Y . Let Y (i) be the set of i-dimensional simplices of Y and let fi (Y ) = |Y (i)|. Let ∆n−1 denote the (n − 1)-dimensional simplex on the vertex set V = [n]. (d−1) For d ≥ 2 let Yd (n, p) denote the probability space of complexes ∆n−1 ⊂ (d) Y ⊂ ∆n−1 with probability measure n Pr(Y ) = pfd (Y ) (1 − p)(d+1)−fd (Y ) .
Let F be an arbitrary fixed field and let Hi (Y ) = Hi (Y ; F) and H i (Y ) = H i (Y ; F) denote the i-th homology and cohomology groups of Y with coefficients in F. Let βi (Y ) = dimF Hi (Y ) = dimF H i (Y ). Kozlov [4] proved the following Theorem 1.1 (Kozlov). For any function ω(n) that tends to infinity ( 1 p = ω(n) n lim Pr [ Y ∈ Yd (n, p) : Hd (Y ) 6= 0 ] = 1 0 p = ω(n)n . n→∞ It is easy to see that if np is bounded away from zero, then the probability that Y ∈ Yd (n, p) contains the boundary of a (d + 1)-simplex does not tend to zero. Thus, the second part of the above statement cannot be improved. Concerning the first part of the statement, as was already observed by Cohen, Costa, Farber and Kappeler for d = 2 (Theorem 6 in [3]), a simple Euler characteristic argument shows that if p = nc where c > d + 1 then a.a.s. Hd (Y ) 6= 0. Our first result is a further improvement on the upper bound in Theorem 1.1. Let gd (x) = (d + 1)(x + 1)e−x + x(1 − e−x )d+1 2
and let cd denote the unique positive solution of the equation gd (x) = d + 1. It is easy to check that gd (d+1) > d+1 and so cd < d+1. A direct calculation yields that cd = d + 1 − Θ( edd ). Theorem 1.2. For a fixed c > cd c lim Pr [Y ∈ Yd (n, ) : Hd (Y ) 6= 0] = 1 . (2) n→∞ n Remark: In the 2-dimensional case, Theorem 1.2 implies that if c > c2 ' 2.783 then Y ∈ Y2 (n, nc ) a.a.s. satisfies H2 (Y ) 6= 0. Simulations indicate that the actual threshold is somewhat lower (around 2.75). We next turn to collapsibility. A (d − 1)−dimensional simplex τ ∈ (d) ∆n−1 (d − 1) is a free face of a complex Y ⊂ ∆n−1 if it is contained in a unique σ ∈ Y (d). Let R(Y ) denote the complex obtained by removing all free (d − 1)-faces of Y together with the d-simplices that contain them. We say that R(Y ) is obtained from Y by a d-collapse step. Let R0 (Y ) = Y and for i ≥ 1 let Ri (Y ) = R(Ri−1 (Y )). We say that Y is d-collapsible if dim R∞ (Y ) < d. Cohen, Costa, Farber and Kappeler [3] proved the following Theorem 1.3 (Cohen, Costa, Farber and Kappeler). If ω(n) → ∞ then 1 ) is a.a.s. 2-collapsible. Y ∈ Y2 (n, ω(n)n Our second result refines Theorem 1.3 and the lower bound in Theorem 1.1 as follows. Let ud (γ, x) = exp(−γ(1 − x)d ) − x . For small positive γ, the only solution of ud (γ, x) = 0 is x = 1. Let γd be the infimum of the set of all nonnegative γ’s for which the equation ud (γ, x) = 0 has a solution x < 1. More explicitly, γd = (dx(1 − x)d−1 )−1 where x satisfies exp(− 1−x ) = x. It is not hard to verify that this yields dx γd = log d + O(log log d). (d−1)
(d)
For ∆n−1 ⊂ Y ⊂ ∆n−1 let s(Y ) denote the number of (d + 1)-simplices in ∆n−1 whose boundary is contained in Y . If c > 0 is fixed and p = nc then a straightforward application of the method of moments (see e.g. Theorem 8.3.1 in [2]) shows that s(Y ) is asymptotically Poisson with parameter c d+2 cd+2 n = . λ = lim E[s] = lim n→∞ n→∞ d + 2 n (d + 2)! 3
The next result asserts that if c < γd and p = nc then s(Y ) > 0 is a.a.s. the only obstruction for d-collapsibility. Let Fn,d denote the family of all (d−1) (d) ∆n−1 ⊂ Y ⊂ ∆n−1 such that s(Y ) = 0. Theorem 1.4. Let c < γd be fixed. Then in the probability space Yd (n, nc ) lim Pr [Y is d − collapsible | Y ∈ Fn,d ] = 1 .
(3)
n→∞
Remark: We have calculated γ2 ' 2.455, and computer simulations suggest that this is indeed the actual threshold for collapsibility for random complexes in Fn,2 . Also, γ3 ' 3.089, γ4 ' 3.508 and γ100 ' 7.555. Clearly, if Y is d-collapsible then Y is homotopy equivalent to a (d − 1)dimensional complex, and in particular Hd (Y ) = 0. Hence for a fixed c < γd and p = nc the following hold: lim Pr [Hd (Y ) = 0 | Y ∈ Fn,d ] = 1
n→∞
and lim Pr [Hd (Y ) = 0] = lim Pr[Fn,d ] = exp(−λ) = exp(−
n→∞
n→∞
cd+2 ). (d + 2)!
The paper is organized as follows. In Section 2 we prove Theorem 1.2. In Section 3 we analyze a random d-tree process that underlies our proof that for c < γd , a random Y ∈ Yd (n, nc ) ∩ Fn,d is a.a.s. d-collapsible. Another main ingredient of the proof is an upper bound on the number of minimal non d-collapsible complexes given in Section 4. In Section 5 we combine these results to derive Theorem 1.4. We conclude in Section 6 with some comments and open problems.
2
The Upper Bound
Let Y ∈ Yd (n, p). Then βi (Y ) = 0 for 0 < i < d − 1 and fi (Y ) = 0 ≤ i ≤ d − 1. The Euler-Poincar´e relation X X (−1)i fi (Y ) = (−1)i βi (Y ) i≥0
i≥0
4
n i+1
for
therefore implies n−1 βd (Y ) = fd (Y ) − + βd−1 (Y ). (4) d The inequality βd (Y ) ≥ fd (Y ) − n−1 already implies that if c > d + 1 d then a.a.s. βd (Y ) 6= 0. As mentioned above, this was observed in the 2dimensional case by Cohen, Costa, Farber and Kappeler [3]. The idea of the proof of Theorem 1.2 is to improve this estimate by providing a non-trivial lower bound on E[βd−1 ]. For τ ∈ ∆n−1 (d − 1) let degY (τ ) = |{σ ∈ Y (d) : τ ⊂ σ}| and let Aτ = {Y ∈ Yd (n, p) : degY (τ ) = 0} . For σ ∈ Y (d) let Lσ be the subcomplex of σ (d−1) given by Lσ = σ (d−2) ∪ {τ ∈ σ (d−1) : degY (τ ) > 1} . Let Pn,d denote the family of all pairs (σ, L), such that σ ∈ ∆n−1 (d) and σ (d−2) ⊂ L ⊂ σ (d−1) . For (σ, L) ∈ Pn,d let Bσ,L = {Y ∈ Yd (n, p) : σ ∈ Y , Lσ = L} . The space of i-cocycles of a complex K is as usual denoted by Z i (K). The space of relative i-cocycles of a pair K 0 ⊂ K is denoted by Z i (K, K 0 ) and will be identified with the subspace of i-cocycles of K that vanish on K 0 . Let z i (K) = dim Z i (K) and z i (K, K 0 ) = dim Z i (K, K 0 ). For a (d − 1)-simplex τ = [v0 , . . . , vd−1 ], let 1τ be the indicator (d − 1)-cochain of τ (i.e. 1τ (η) = sgn(π) if η = [vπ(0) , . . . , vπ(d−1) ] and is zero otherwise). If τ ∈ ∆n−1 (d − 1) then Z d−1 (τ ) is the 1-dimensional space spanned by 1τ . If (σ, L) ∈ Pn,d and fd−1 (L) = j, then z d−1 (σ, L) = d − j. indeed, suppose σ = [v0 , . . . , vd ] and for 0 ≤ i ≤ d let τi = [v0 , . . . , vbi , . . . , vd ]. If L(d − 1) = {τi }di=d−j+1 then d−1 {1τ0 − 1τi }d−j (σ, L) . i=1 forms a basis of Z Claim 2.1. For any Y ∈ Yd (n, p) M Z d−1 (Y ) ⊃ Z d−1 (τ ) ⊕ {τ ∈∆n−1 (d−1) : Y ∈Aτ }
M {(σ,L)∈Pn,d : Y ∈Bσ,L }
5
Z d−1 (σ, L) .
Proof: The containment is clear. To show that the right hand side is a direct sum note that nontrivial cocycles in different summands must have disjoint supports and are therefore linearly independent. Let a(Y ) = |{τ ∈ ∆n−1 (d − 1) : degY (τ ) = 0}| and for 0 ≤ j ≤ d let αj (Y ) = |{(σ, L) ∈ Pn,d : Y ∈ Bσ,L , fd−1 (L) = j}|. Note that αj (Y ) is the number of d-faces of Y that contain exactly d + 1 − j (d − 1)-faces of degree 1. By Claim 2.1 z
d−1
def
(Y ) ≥ u(Y ) = a(Y ) +
d X
αj (Y )(d − j) .
(5)
j=0 n−1 d−1
, it follows from (4) and (5) n def βd (Y ) ≥ v(Y ) = fd (Y ) + u(Y ) − . (6) d Theorem 1.2 will thus follow from As βd−1 (Y ) = dim H d−1 (Y ) = z d−1 (Y ) − that
Theorem 2.2. Let c > cd and let p = nc . Then lim Pr [ Y ∈ Yd (n, p) : v(Y ) ≤ 0] = 0 .
n→∞
Proof: First note that
n c E[fd ] = p= nd − O(nd−1 ) , d+1 (d + 1)! n e−c d E[a] = (1 − p)n−d = n − O(nd−1 ) , d d! and for 0 ≤ j ≤ d
n d+1 E[αj ] = p(1 − p)(n−d−1)(d+1−j) (1 − (1 − p)n−d−1 )j d+1 j nd c d + 1 −c(d+1−j) = e (1 − e−c )j − O(nd−1 ) . (d + 1)! j 6
Therefore E[u] = E[a] +
d X
E[αj ](d − j)
j=0
d nd e−c nd c X d + 1 −c(d+1−j) = + e (1 − e−c )j (d − j) − O(nd−1 ) d! (d + 1)! j=0 j =
nd ((1 + c)(d + 1)e−c − c(1 − (1 − e−c )d+1 )) − O(nd−1 ). (d + 1)!
It follows that n E[v] = E[fd ] + E[u] − d d n (c + (1 + c)(d + 1)e−c − c(1 − (1 − e−c )d+1 ) − (d + 1)) − O(nd−1 ) = (d + 1)! nd = (gd (c) − d − 1) − O(nd−1 ) . (d + 1)! Since c > cd it follows that for sufficiently large n E[v] ≥ nd
(7)
where > 0 depends only on c and d. To show that v is a.a.s. positive we use the following consequence of Azuma’s inequality due to McDiarmid [6]. Theorem 2.3. Suppose f : {0, 1}m → R satisfies |f (x) − f (x0 )| ≤ T if x and x0 differ in at most one coordinate. Let ξ1 , . . . , ξm be independent 0, 1 valued random variables and let F = f (ξ1 , . . . , ξm ). Then for all λ > 0 Pr[F ≤ E[F ] − λ] ≤ exp(−
2λ2 ). T 2m
(8)
n Let m = d+1 and let σ1 , . . . , σm be an arbitrary ordering of the dsimplices of ∆n−1 . Identify Y ∈ Yd (n, p) with its indicator vector (ξ1 , . . . , ξm ) where ξi (Y ) = 1 if σi ∈ Y and ξi (Y ) = 0 otherwise. Note that if Y and Y 0 differ in at most one d-simplex then |a(Y ) − a(Y 0 )| ≤ d + 1 and |αj (Y ) − αj (Y 0 )| ≤ d + 1 for all 0 ≤ j ≤ d. It follows that |v(Y ) − v(Y 0 )| ≤ T = 2d3 .
7
Applying McDiarmid’s inequality (8) with F = v and λ = E[v] it follows that 2E[v]2 Pr[v ≤ 0] ≤ exp(− 3 2 ) ≤ exp(−C2 nd−1 ) (2d ) m for some C2 = C2 (c, d) > 0.
Remark: The approach used in the proof of Theorem 1.2 can be extended as d−1 follows. For a fixed `, let Z(`) (Y ) ⊂ Z d−1 (Y ) denote the subspace spanned by (d − 1)-cocycles φ ∈ Z d−1 (Y ) such that |supp(φ)| ≤ `. Let d−1 E[dim Z(`) (Y )] θd,` (x) = lim n n→∞
d
where the expectation is taken in the probability space Yd (n, nx ). For example, it was shown in the proof of Theorem 1.2 that θd,1 (x) = e−x and θd,2 (x) = (1 + x)e−x −
x (1 − (1 − e−x )d+1 ). d+1
Let x = cd,` denote the unique positive root of the equation x + (d + 1)θd,` (x) = d + 1. The following fact is implicit in the proof of Theorem 1.2. Proposition 2.4. For any fixed c > cd,` c lim Pr [Y ∈ Yd (n, ) : Hd (Y ) 6= 0] = 1 . n→∞ n
(9)
Let c˜d = lim`→∞ cd,` . It seems likely that c˜nd is the exact threshold for the vanishing of H d (Y ). This is indeed true in the graphical case d = 1. Proposition 2.5. c˜1 = 1.
8
Proof: For a subtree K = (VK , EK ) on the vertex set VK ⊂ [n] let AK denote all graphs G ∈ G(n, p) that contain K as an induced subgraph and contain no edges in the cut (VK , VK ). The space of 0-cocycles Z 0 (K) is 1-dimensional and is spanned by the indicator function of VK . As in Claim 2.1 it is clear that for G ∈ G(n, p) and a fixed ` M Z 0 (G) ⊃ Z 0 (K). {K:|VK |≤` and G∈AK }
Hence for p =
x n
0 E[dim Z(`) (G)] =
X {Pr[AK ] : K is a tree on ≤ ` vertices} =
` X k−1 n k−2 x k−1 x k ( ) (1 − )k(n−k)+( 2 ) ∼ n n k k=1
n
` X k k−2 k=1
P∞
kk−2
Let S(z) = k=1 ber of trees. Then
k!
k!
`
x
k−1 −xk
e
z k be the exponential generating function for the num-
lim θ1,` (x) = lim
`→∞
n X k k−2 = (xe−x )k . x k=1 k!
0 (G)] E[dim Z(`)
n
`,n→∞
=
S(xe−x ) . x
Therefore c˜1 = lim`→∞ c1,` is the solution of the equation x+
2S(xe−x ) = 2. x
(10)
P∞ kk−1 k Let R(z) = k=1 k! z be the exponential generating function for the number of rooted trees. It is classically known (see e.g. [7]) that R(z) = z exp(R(z)), and that S(z) = R(z) − 21 R(z)2 . It follows that R(e−1 ) = 1 and S(e−1 ) = 21 . Hence c˜1 = 1 is the unique solution of (10).
9
3
The Random d-Tree Process
A simplicial complex T on the vertex set V with |V | = ` ≥ d is a d-tree if there exists an ordering V = {v1 , . . . , v` } such that lk(T [v1 , . . . , vi ], vi ) is a (d − 1)-dimensional simplex for all d + 1 ≤ i ≤ `. Let GT denote the graph with vertex set T (d − 1), whose edges are the pairs {τ1 , τ2 } such that τ1 ∪ τ2 ∈ T (d). Let distT (τ1 , τ2 ) denote the distance between τ1 and τ2 in the graph GT . A rooted d-tree is a pair (T, τ ) where T is a d-tree and τ is some (d−1)-face of T . Let τ be a fixed (d − 1)-simplex. Given k ≥ 0 and γ > 0 we describe a random process that gives rise to a probability space Td (k, λ) of all d-trees T rooted at τ such that distT (τ, τ 0 ) ≤ k for all τ 0 ∈ T . The definition of Td (k, λ) proceeds by induction on k. Td (0, γ) is the (d − 1)-simplex τ . Let k ≥ 0. A d-tree in Td (k + 1, γ) is generated as follows: First generate a T ∈ Td (k, λ) and let U denote all τ 0 ∈ T (d − 1) such that distT (τ, τ 0 ) = k. Then, independently for each τ 0 ∈ U, pick J = Jτ 0 new vertices z1 , . . . , zJ where J is Poisson distributed with parameter γ, and add the d-simplices z1 τ 0 , . . . , zJ τ 0 to T . We next define the operation of pruning of a rooted d-tree (T, τ ). Let {τ1 , . . . , τ` } be the set of all free (d−1)-faces of T that are distinct from τ , and let σi be the unique d-simplex of T that contains τi . The d-tree T 0 obtained from T by removing the simplices τ1 , σ1 , . . . , τ` , σ` is called the pruning of T . Clearly, any T ∈ Td (k +1, γ) collapses to its root τ after at most k +1 pruning steps. Denote by Cd (k + 1, γ) the event that T ∈ Td (k + 1, γ) collapses to τ after at most k pruning steps, and let ρd (k, γ) = Pr[Cd (k + 1, γ)]. Clearly, ρd (0, γ) is the probability that T ∈ Td (1, γ) consists only of τ , hence ρd (0, γ) = e−γ .
(11)
Let σ1 , . . . , σj denote the d-simplices of T ∈ Td (k + 1, γ) that contain τ and for each 1 ≤ i ≤ j let ηi1 , . . . , ηid be the (d − 1)-faces of σi that are different from τ . Let Ti` ∈ Td (k, λ) denote the subtree of T that grows out of ηi` . Clearly, T collapses to τ after at most k pruning steps iff for each 1 ≤ i ≤ j, at least one of the d-trees Ti` collapses to its root ηi` in at most k − 1 steps.
10
We therefore obtain the following recursion: ρd (k, γ) = =
∞ X j=0 ∞ X j=0
Pr [J = j](1 − (1 − ρd (k − 1, γ))d )j γ j −γ e (1 − (1 − ρd (k − 1, γ))d )j j!
(12)
= exp(−γ(1 − ρd (k − 1, γ))d ) . Equations (11) and (12) imply that the sequence {ρd (k, γ)}k is non-decreasing and converges to ρd (γ) ∈ (0, 1], where ρd (γ) is the smallest positive solution of the equation ud (γ, x) = exp(−γ(1 − x)d ) − x = 0. (13) If γ ≥ 0 is small, then ρd (γ) = 1. Let γd denote the infimum of the set of nonnegative γ’s for which ρd (γ) < 1. The pair (γ, x) = (γd , ρd (γd )) satisfies both d (γd , ρd (γd )) = 0. A straightforward computation ud (γd , ρd (γd )) = 0 and ∂u ∂x shows that γd = (dx(1 − x)d−1 )−1 where x = ρd (γd ) is the unique solution of exp(− 1−x ) = x. dx
4
The Number of Non-d-Collapsible Complexes
When we discuss d-collapsibility, we only care about the inclusion relation between d-faces and (d − 1)-faces. Therefore, in the present section we can and will simplify matters and consider only the complex that is induced from our (random) choice of d-faces. Namely, for every i ≤ d, a given i-dimensional face belongs to the complex iff it is contained in some of the chosen d-faces. A complex is a core if every (d − 1)-dimensional face belongs to at least two simplices, so that not even a single collapse step is possible. A core complex is called a minimal core complex if none of its proper subcomplexes is a core. The main goal of this section is to show that with almost certainty there are just two types of minimal core subcomplexes that a sparse random complex can have. It can either be the boundary of a (d + 1)-simplex, ∂∆d+1 , or it must be very large. Obviously this implies that there are no small noncollapsible subcomplexes which do not contain the boundary of a (d + 1)simplex.
11
Theorem 4.1. For every c > 0 there exists a constant δ = δ(c) > 0 such that a.a.s. every minimal core subcomplex K of Y ∈ Yd (n, nc ) with fd (K) ≤ δnd , must contain the boundary of a (d + 1)-simplex. Henceforth we use the convention that faces refer to arbitrary dimensions, but unless otherwise specified, the word simplex is reserved to mean a d-face. Our proof uses the first moment method. In the main step of the proof we obtain an upper bound on Cd (n, m), the number of all minimal core ddimensional complexes on vertex set [n] = {1, 2, . . . , n}, which contain m simplices. Two simplices are considered adjacent if their intersection is a (d−1)- face. · If A ∪ B is a splitting of a minimal core complex, then there is a simplex in A and one in B that are adjacent, otherwise the corresponding subcomplexes are cores as well. Therefore K can be constructed by successively adding a simplex that is adjacent to an already existing simplex. This consideration easily yields an upper bound of nd+m on Cd (n, m). The point is that if m = δnd for δ > 0 small enough, we get an exponentially smaller (in m) upper bound and this is crucial for our analysis. Lemma 1. Let m = δnd and δ > 0 small enough. Then 1 nd−1 (14) nd nm (2d+1 d3 δ d4 )m . Cd (n, m) ≤ d−1 2 (d m) d 1 d(d+1)δ d Proof. Let b = . A (d − 2)-face is considered heavy or light de2 pending on whether it is covered by at least bn (d − 1)-faces or less. The sets of heavy and light (d − 2)-faces are denoted by Hd−2 and Ld−2 repectively. We claim that |Hd−2 | ≤ bd−1 nd−1 . To see this note that each simplex contains exactly d + 1 (d − 1)-faces, but the complex is a core, so that each (d − 1)-face is covered at least twice. Consequently, our complex has at most m(d+1) (d − 1)-faces. Likewise, each (d − 1)-face contains d (d − 2)-faces. Each 2 heavy (d − 2)-face is covered at least bn times and the claim follows by the following calculation:
|Hd−2 | ≤
d(d + 1)m d(d + 1)δnd d(d + 1)δnd−1 = = = bd−1 nd−1 . 2bn 2bn 2b
We extend the heavy/light dichotomy to lower dimensions as well. For each 0 ≤ i ≤ d − 3, an i-face is considered heavy if it covered by at least b · n 12
heavy (i + 1)-faces. Otherwise it is light. The sets of heavy/light i-faces are denoted by Hi resp. Li . By counting inclusion relations between heavy faces i+1 | which yields of consecutive dimensions it is easily seen that |Hi | ≤ (i+2)|H bn |Hi | ≤
(d − 1)! i+1 i+1 b n . (i + 1)!
The set of i-dimensional heavy (resp. light) faces contained in a given face σ is denoted by Hiσ be (resp. Lσi ). The bulk of the proof considers a sequence of complexes C1 , . . . , Cm = C, where the complex Ci is obtained from Ci−1 by adding a single simplex. A (d−1)-face σ of Ci can be saturated or unsaturated. This depends on whether or not every simplex in Cm that contains σ already belongs to Ci . Prior to defining the complexes Ci , we specify the set of heavy (d − 2)-faces in one of n at most (d−1) possible ways. Note that this choice uniquely determines bd−1 nd−1
the sets of heavy faces for every dimension 0 ≤ i ≤ d − 3. We start off n with the complex C1 , which has exactly one simplex. Clearly there are d+1 possible choices for C1 . We move from Ci−1 to Ci by adding a single simplex ti , which covers a chosen unsaturated (d − 1)-face σi−1 of Ci−1 . Our choices are subject to the condition that every heavy (d − 2)-face in Cm is one of the heavy (d − 2)-faces chosen prior to the process. In other words, we must never make choices that create any additional heavy faces in addition to those derived from our preliminary choice. Our goal is to bound the number of choices for this process. The crux of the argument is a rule for selecting the chosen face. Associated with every face is a vector counting the number of its heavy vertices, its heavy edges, its heavy 2-faces etc. The chosen face is always lexicographically minimal w.r.t. this vector, breaking ties arbitrarily. A (d − 1)-face all of whose subfaces are light is called primary. In each step j we expand a (d − 1)-face σ to a simplex σ ∪ y. Such a step is called a saving step if either: 1. The vertex y is heavy. 2. There exists a light (d − 2)- subface τ ⊂ σ such that τ ∪ y is contained in a simplex in Cj−1 . 3. There exists a light subface τ ⊂ σ such that the face τ ∪ y is heavy.
13
Note that the number of choices of y in the first case is ≤ |H0 | ≤ (d − 1)! · b · n. In the second case the number of choices for y is at most dbn. In the third case there are d − 2 possibilities for the dimension of the d light face and for each such dimension i there are at most i+1 bn choices for y. In all cases the number of choices for y is at most ≤ dd bn. A step that is not saving is considered wasteful. For wasteful steps we bound the number of choices for y by n. The idea of the proof is that every such a process which produces a minimal core complex must include many saving steps. More specifically, we want to show: Claim 4.2. For every d3 wasteful steps, at least one saving step is carried out. Proof. The proof of this claim consists of two steps. We show that there is no sequence of d(d − 1) consecutive wasteful steps, without the creation of an unsaturated primary face. Also, the creation of d + 1 primary faces necessarily involves a saving step. If u is a vertex in a (d − 1)-face σ, let riσ (u) be the number of heavy i-faces in σ that contain u. Also, Viσ denotes the set of vertices v in σ that are included only in light i-subfaces of σ. Proposition 4.3. Let σ and σ 0 be two consecutively chosen faces where σ is non-primary and the extension step on σ is wasteful. Then σ 0 precedes σ in 0 the order of faces and |Viσ | ≥ |Viσ | + 1 where i is the smallest dimension for which |Hiσ | > 0. Proof. Since the extension step on σ is wasteful (and, in particular, not a saving step of type (iii)) and since all j-subfaces of σ are light for j < i, every j-face in σ ∪ y is light. Moreover, every i-subface of σ ∪ y that contains y is light as well. We claim that σ 0 = σ \{u}∪{y} where the vertex u of σ maximizes riσ (v). (Since |Hiσ | > 0, there are vertices v in σ for which riσ (v) > 0). Notice that σ has d−1 −riσ (u) more light i-subfaces than does τ u := σ\u. i u Namely, |Lτi | = |Lσi | − d−1 + riσ (u). i Combining the fact that every i-subface of σ ∪ y that contains y is light τyu u u u we see that in τy := τ ∪ y, |Li | = |Lτi | + d−1 = |Lσi | + riσ (u). But since i σ u u ri (u) > 0, τy precedes σ. In this case τy must be a new face that does not 14
belong to the previous complex, or else it would have been preferred over σ. Being a new face, it is necessarily unsaturated. Since u maximizes riσ (u) over all vertices in σ, it follows that τyu precedes all other faces created in the expansion. Furthermore, no other face precedes σ or else it would be chosen 0 rather than σ. Thus σ 0 = τuy , as claimed. Notice that y ∈ Viσ and also 0 Viσ ⊆ Viσ (Note that every i-dimensional subfaces of σ 0 that is not contained 0 in σ is light since it contains y). Thus |Viσ | ≥ |Viσ | + 1. Consider a chosen non-primary face σ and let i be the smallest dimension for which |Hiσ | > 0. The previous claim implies that after at most d consecutive wasteful steps the chosen face, θ precedes σ and |Viθ | = d. Then |Hjθ | = 0 for all j ≤ i (in particular |Hiθ | = 0). By repeating this argument d − 1 times we conclude that following every series of d(d − 1) consecutive wasteful steps, a primary face must be chosen: After at most d consecutive wasteful steps the chosen face can have no heavy vertices. At the end of the next d consecutive wasteful steps, the chosen face has no heavy vertices nor heavy edges. Repeating this argument (d − 1) times necessarily leads us to a chosen primary face. Proposition 4.4. Only saving steps can decrease the number of unsaturated primary faces. Proof. Let σ = a1 , a2 . . . , ad be a primary face and let y be the vertex that expands it. Denote the (d − 1)-face {a1 , a2 . . . , ai−1 , ai+1 , . . . , ad } ∪ {y} by σ i . Since this is not a saving step of type (i), y is light. It is also not of type i (iii) and so |Hkσ | = 0 for every i = 1, . . . , d and k = 1, . . . , d − 2, so that faces σ i are primary. However this is not a type (ii) saving step, so all the (d − 1)-faces σ i must be new. Thus the number of unsaturated primary faces has increased by at least d − 1. The proof of Claim 4.2 is now complete, since at each step at most d + 1 faces get covered. We can turn now to bound the number of minimal core m-simplex complexes Cm . As mentioned, we first specify the heavy (d − 2)-faces of Cm by n d−1 d−1 specifying a set of b n out of the total of d−1 (d − 2)-faces. Then we select the first simplex C1 and mark all its (d − 1)-faces as unsaturated. In order to choose the i-th step we first decide whether it is a saving or wasteful step, and if it is a saving step, what type it has. There is a total of d + 1 possible kinds of extensions of the current (d − 1)-face: A saving step of type 15
(i), (ii), or one of the d − 2 choices of type (iii) (according to dimension), or a wasteful step. In a saving step the expanding vertex can be chosen in at most dd bn ways. The number of possible extension clearly never exceeds n and it is this trivial upper bound that we use for wasteful steps. Finally we update the labels on the (d − 1)-faces of a new simplex. We need to decide which of the unsaturated (d − 1)-faces that are already covered by at least two simplices change their status to saturated. There are at most 2d+1 possibilities of such an update. As we saw, at least dm3 of the steps in such process are saving steps. Consequently we get the following upper bound on Cd (n, m), the number of minimal core n-vertex d-dimensional complexes with m simplices. (In reading the expression below, note that the terms therein correspond in a one-to-one manner to the ingredients that were just listed).
n d−1 nd+1 (d bd−1 nd−1
m
+ 1)m−1 nm−1 (dd b) d3 (2d+1 )m nd−1 d m d+1 d12 d13 m n n ((d + 1)2 d b ) ≤ d−1 (d2 δ) d nd−1 1 nd−1 nd−1 d m d+1 2 2 d14 m ≤ n n (2 d (d δ) ) ≤ nd nm (2d+1 d3 δ d4 )m . d−1 d−1 2 2 (d m) d (d m) d
Proof of Theorem 4.1: We show the assertion with δ = δ(c) = 4 (2d+2 d3 c)−d . Indeed, consider a complex drawn from Yd (n, p). Let Xm = Xm (n, p) count the number of minimal core subcomplexes with m simplices and which are not copies of ∂∆d+1 . Our argument splits according to whether m is small or large, the dividing line being m = m1 = (d3 log n)d . The theorem speaks only about the range m ≤ m2 = δ(c)nd . By Lemma 1,
16
m2 X
EXm ≤
m
Cd (n, m)p ≤
m=m1
m=m1
≤ nd
m2 X
m2 X
d−1 2 m) d
(nd−1 )(d
1
nd 2d+1 d3 δ d4 c
m
m=m1
d−1 2 m2 1 (d m) d d−1 X 2 d m (d m) d nd−1 nd−1 2− d2 2−m ≤ nd
m2 X m=m1
m=m1
m2 X 1 ≤n n m=m
d−1 (d2 m) d
d
= o(1)
1
It follows that with almost certainty no cores with m simplices occur where m2 ≥ m ≥ m1 . We next consider the range d + 3 ≤ m ≤ m1 . Note d+3 that a minimal core complex with m ≥ d + 3 simplices has at most d+4 m vertices. Let ∆(u) denote the number of simplices that contain the vertex u. Clearly, if ∆(u) > 0 then ∆(u) ≥ d + 1 (Consider a simplex σ that contains u. Every face of the form σ \ w with u 6= w ∈ σ is covered by a simplex other than σ). It is not hard to verify that if some simplex σ contains two distinct vertices with ∆(u) = ∆(w) = d + 1 then the complex contains ∂∆d+1 contrary to the minimality assumption. Let t be the number of vertices u m . with ∆(u) = d + 1. No simplex contains two such vertices, so that t ≤ d+1 Counting vertices in the complex according to the value of ∆ we get (d + 1)t + (d + 2)(v − t) ≤ (d + 1)m where v is the total number of vertices. The conclusion follows. The expected number of minimal core subcomplexes of Yd (n, p) that contain d + 3 ≤ m ≤ m1 simplices satisfies h m1 m1 m1 h im X X X d+3 d + 3 d+1 im n m d+1 d+4 m p EXm ≤ m p ≤ n d+3 m d+4 d+4 m=d+3 m=d+3 m=d+3 ≤
m1 X m=d+3
cd+4
m log2d(d+1)(d+4) n d+4 = o(1). n
Consequently, a.a.s. Yd (n, p) contains no minimal core subcomplexes of m 4 simplices with d + 3 ≤ m ≤ (2d+2 d3 c)−d nd . 17
5
The Threshold for d-Collapsibility (d)
For a complex Y ⊂ ∆n−1 and a fixed τ ∈ ∆n−1 (d − 1), define a sequence of complexes {Si (Y )}i≥0 as follows. S0 (Y ) = τ and for i ≥ 1 let Si (Y ) be the union of Si−1 (Y ) and the complex generated by all the d-simplices of Y that contain some η ∈ Si−1 (Y )(d − 1). Let Td denote the family of all d-trees. Consider the events Ak , D ⊂ Yd (n, p) given by Ak = {Sk (Y ) ∈ Td } and D = {degY (η) ≤ log n for all η ∈ ∆n−1 (d − 1)}. Claim 5.1. Let k and c > 0 be fixed and p = nc . Then Pr[Ak+1 ∩ D] = 1 − o(1). Proof: Fix η ∈ ∆n−1 (d − 1). The random variable degY (η) has a binomial distribution B(n − d, nc ), hence by the large deviations estimate Pr[degY (η) > log n] < n−Ω(log log n) . Therefore Pr[D] = 1 − o(1). If Y ∈ D then f0 (Sk+1 (Y )) = O(logk+1 n) and fd−1 (Sk (Y )) = O(logk n). Note that Sk+1 (Y ) is a d-tree iff in its generation process, we never add a simplex of the form ηv such that both η ∈ ∆n−1 (d−1) and v ∈ ∆n−1 (0) already exist in the complex. Since the number of such pairs is at most f0 (Sk+1 (Y ))fd−1 (Sk (Y )) it follows that Pr[Ak+1 ∩ D] ≥ (1 − (1 −
c f0 (Sk+1 (Y ))fd−1 (Sk (Y )) ) ≥ n
c O(log2k+1 n) ) = 1 − o(1) . n
(d)
For Y ⊂ ∆n−1 let r(Y ) = fd (R∞ (Y )) be the number of d-simplices remaining in Y after performing all possible d-collapsing steps. For τ ∈ ∆n−1 (d − 1) let Γ(τ ) = {σ ∈ ∆n−1 (d) : σ ⊃ τ }. Claim 5.2. Let 0 < c < γd be fixed and p = nc . Then for any fixed τ ∈ ∆n−1 (d): Pr[R∞ (Y ) ∩ Γ(τ ) 6= ∅] = o(1) . (15) 18
Proof: Let δ > 0. Since c < γd lim ρd (k, c) = ρd (c) = 1.
k→∞
Choose a fixed k such that δ ρd (k, c) > 1 − . 3 Claim 5.1 implies that if n is sufficiently large then δ Pr[Ak+1 ∩ D] ≥ 1 − . 3 Next note that if Y ∈ Ak+1 then Sk+1 = Sk+1 (Y ) can be generated by the following inductively defined random process: S0 = τ . Let 0 ≤ i ≤ k. First generate T = Si and let U denote all τ 0 ∈ T (d − 1) such that distT (τ, τ 0 ) = i. Then, according to (say) the lexicographic order on U, for each τ 0 ∈ U pick J new vertices z1 , . . . , zJ according to the binomial distribution B(n − n0 , nc ), where n0 is the number of vertices that appeared up to that point, and add the d-simplices z1 τ 0 , . . . , zJ τ 0 to T . Note that the process described above is identical to the d-tree process of Section 3, except for the use of the binomial distribution B(n − n0 , nc ) instead of the Poisson distribution P o(c). Now if Y ∈ Ak+1 ∩ D then n0 = O(logk+1 n) at all stages of this process. It follows that if n is sufficiently large then the total variation distance between the distributions Sk+1 (Y ) and Td (k+1, c) is less then 3δ . Denote by Cd0 (k+1, c) the event that Sk+1 (Y ) is in Ak+1 and collapses to τ in at most k pruning steps. The crucial observation now is that if Y ∈ Cd0 (k+1, c) then R∞ (Y )∩Γ(τ ) = ∅. It follows that Pr[R∞ (Y ) ∩ Γ(τ ) 6= ∅] ≤ (1 − Pr[Ak+1 ∩ D]) + Pr[Y 6∈ Cd0 (k + 1, c)] ≤ (1 − Pr[Ak+1 ∩ D]) + dT V (Sk+1 (Y ), Td (k + 1, c)) + (1 − Pr[Cd (k + 1, c)]) ≤ δ δ δ + + = δ. 3 3 3 Let G(Y ) = {τ ∈ ∆n−1 (d − 1) : R∞ (Y ) ∩ Γ(τ ) 6= ∅} and let g(Y ) = |G(Y )|. For a family G ⊂ ∆n−1 (d − 1) let w(G) denote the set of all d-simplices σ ∈ ∆n−1 (d) all of whose (d − 1)-faces are contained in G. Using Claim 5.2 we establish the following 19
Theorem 5.3. Let δ > 0 and 0 < c < γd be fixed and let p = nc . Then Pr[fd (R∞ (Y )) > δnd ] = o(1). Proof: Let 0 < = (d, c, δ) < 1 be a constant whose value will be fixed later. Clearly Pr[fd (R∞ (Y )) > δnd ] ≤ Pr[g(Y ) > δnd ] + Pr[g(Y ) ≤ δnd
& fd (R∞ (Y )) > δnd ].
To bound the first summand note that E[g] = o(nd ) by Claim 5.2. Hence by Markov’s inequality Pr[g(Y ) > δnd ] ≤ (δnd )−1 E[g] = o(1). Next note that Pr[g(Y ) ≤ δnd X
& fd (R∞ (Y )) > δnd ] ≤ Pr[|w(G) ∩ Y (d)| > δnd ].
{G⊂∆n−1 (d−1):|G|=δnd }
Fix a G ⊂ ∆n−1 (d−1) such that |G| = δnd . By the Kruskal-Katona theorem there exists a C1 = C1 (d, δ) such that N = |w(G)| ≤ C1
d+1 d
nd+1 .
Applying the large deviation estimate for the binomial distribution B(N, nc ) 1 and writing C2 = ecC we obtain δ Pr[|w(G) ∩ Y (d)| > δnd ] ≤ (C2 On the other hand
n d
δnd Choosing such that (
≤(
d+1 d
d
)δn .
e δnd ) . δ
d+1 e ) C2 d < e−1 δ
it follows that Pr[g(Y ) ≤ δnd
& fd (R∞ (Y )) > δnd ] ≤ exp(−δnd ). 20
Proof of Theorem 1.4: Let c < γd and p = nc . By Theorem 4.1 there exists a δ > 0 such that a.a.s. any non-d-collapsible subcomplex K of Y ∈ Yd (n, nc ) such that fd (K) ≤ δnd contains the boundary of a (d + 1)-simplex. It follows that Pr [Y non d − collapsible | Y ∈ Fn,d ] = Pr [fd (R∞ (Y )) > 0 | Y ∈ Fn,d ] ≤ Pr [fd (R∞ (Y )) > δnd ]·Pr [ Y ∈ Fn,d ]−1 +Pr [0 < fd (R∞ (Y )) ≤ δnd | Y ∈ Fn,d ]. The first summand is o(1) by Theorem 5.3, and the second summand is o(1) by Theorem 4.1.
6
Concluding Remarks
Let us remark that one may show a random process statement slightly stronger than Theorem 4.1 (see [5], where a similar result is shown for the k-core of random graphs). More specifically, let us define the d-dimensional (n) random process Yd = {Yd (n, M )}Md+1 =0 as the Markov chain whose stages are simplicial complexes, which starts with the full (d − 1)-dimensional skeleton of ∆n−1 and no d-simplices, and in each stage Yd (n, M + 1) is obtained from Yd (n, M ) by adding to it one d-simplex chosen uniformly at random from all the d-simplices which do not belong to Yd (n, M ). The core of a complex Y is the maximal core subcomplex of Y . Then, the following holds. Theorem 6.1. There exists a constant α = α(d) > 0 such that for almost ev(n) ery d-dimensional random process Yd = {Yd (n, M )}Md+1 =0 there exists a stage ˆ ˆ ˆ M = M (Yd ) such that the core of Yd (n, M ) is of the size O(1) and consists ˆ + 1) contains at of boundaries of (d + 1)-simplices, while the core of Yd (n, M d least αn d-simplices. Many questions remain open. The most obvious ones are • What is the threshold for d-collapsibility of random simplicial complexes in Fn,d ? We conjecture that it is indeed p = γd /n. 21
• Find the exact threshold for the nonvanishing of Hd (Y ). The first two authors (see [1]) have recently improved the upper bound given in Theorem 1.2 and they conjecture that their new bound is in fact sharp. This in particular would imply that the threshold does not depend on the underlying field. • Although this question is implicitly included in the above two questions, it is of substantial interest in its own right: Can you show that the two thresholds (for d-collapsibility and for the vanishing of the top homology) are distinct? We conjecture that the two thresholds are, in fact, quite different. In particular, although d-collapsibility is a sufficient condition for the vanishing of Hd , there is only a vanishingly small probability that a random simplicial complex with trivial top homology is d-collapsible.
References [1] L. Aronshtam and N. Linial, When does the top homology of a random simplicial complex vanish?, arXiv:1203.3312. [2] N. Alon and J. Spencer, The Probabilistic Method, 2nd Edition, WileyIntescience, 2000. [3] D. Cohen, A. Costa, M. Farber and T. Kappeler, Topology of Random 2-Complexes, Discrete Comput. Geom. 47(2012) 117–149. [4] D. Kozlov, The threshold function for vanishing of the top homology group of random d-complexes, Proc. Amer. Math. Soc. 138(2010) 4517– 4527. [5] T. Luczak, Size and connectivity of the k-core of a random graph, Discrete Math. 91(1991) 61–68. [6] C. McDiarmid, On the Method of Bounded Differences, Surveys in combinatorics 1989, 148–188, London Math. Soc. Lecture Note Ser., 141, Cambridge Univ. Press, Cambridge, 1989. [7] R. Stanley, Enumerative Combinatorics Vol. I, Cambridge Unversity Press, Cambridge, 1997.
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